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Theory
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Spectral processing.doc
Category: Signal processing
Topic: Spectral processing
Digital signal processing
Time and frequency domains
It is a property of all real waveforms that they can be made up
of a number of sine waves of certainamplitudes and frequencies.
Viewing these waves in the frequency domain rather than the time
domaincan be useful in that all the components are more readily
revealed.
time frequency
amplitude
Each sine wave in the time domain is represented by one spectral
line in the frequency domain. Theseries of lines describing a
waveform is known as its frequency spectrum.
Fourier transform
The conversion of a time signal to the frequency domain (and its
inverse) is achieved using the FourierTransform as defined
below.
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This function is continuous and in order to use the Fourier
Transform digitally a numerical integrationmust be performed
between fixed limits.
The Discrete Fourier Transform (DFT)
The digital computation of the Fourier Transform is called the
Discrete Fourier Transform. It calculatesthe values at discrete
points (mDf) and performs a numerical integration as illustrated
below betweenfixed limits (N samples).
Since the waveform is being sampled at discrete intervals and
during a finite observation time, we donot have an exact
representation of it in either domain. This gives rise to
shortcomings which arediscussed later.
Hermitian symmetry
The Fourier transform of a sinusoidal function would result in
complex function made up of real andimaginary parts that are
symmetrical. This is illustrated below. In the majority of cases
only the realpart is taken into account and of this only the
positive frequencies are shown. So the representation ofthe
frequency spectrum of the sine wave shown below would become the
area shaded in grey.
+f0-f+f0-f
A
A/2
A/2A/2
S(f) imaj S(f) realX(t)
The Fast Fourier Transform (FFT)
The Fast Fourier Transform is a dedicated algorithm to compute
the DFT. It thus determines thespectral (frequency) contents of a
sampled and discretized time signal. The resulting spectrum is
alsodiscrete. The reverse procedure is referred to as an inverse or
backward FFT.
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time
frequency
N samples
N/2 spectral lines
inverse
To achieve high calculation performance the FFT algorithm
requires that the number of time samples(N) be a power of 2 (such
as 2, 4, 8, ...., 512, 1024, 2048).
Blocksize
Such a time record of N samples is referred to as a block of
data with N being the blocksize. Nsamples in the time domain
converts to N/2 spectral (frequency) lines. Each line contains
informationabout both amplitude and phase.
Frequency range
The time taken to collect the sample block is T. The lowest
frequency that can be detected then is thatwhich is the reciprocal
of the time T.
TThe frequency spacing between the spectral lines is therefore
1/T and the highest frequency that canbe determined is
(N/2).(1/T).
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The frequency range that can be covered is dependant on both the
blocksize (N) and the samplingperiod (T). To cover high frequencies
you need to sample at a fast rate which implies a short
sampleperiod.
Real time Bandwidth
Remember that an FFT requires a complete block of data to be
gathered before it can transform it. Thetime taken to gather a
complete block of data depends on the blocksize and the frequency
range but itis possible to be gathering a second time record while
the first one is being transformed. If thecomputation time takes
less than the measurement time, then it can be ignored and the
process is saidto be operating in real time.
timerecord 1
timerecord 2
timerecord3
timerecord 4
FFT 1 FFT 2 FFT 3
Real timeoperation
timerecord 1
timerecord 2
timerecord3
timerecord 4
FFT 1 FFT 2 FFT 3
This is not the case if the computation time is taking longer
than the measurement time or if theacquisition requires a trigger
condition.
Overlap
Overlap processing involves using time records that are not
completely independent of each other asillustrated below.
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timerecord 1
timerecord 2
timerecord3
timerecord 4
FFT 1 FFT 2 FFT 3
If the time data is not being weighted at all by the application
of a window, then overlap processingdoes not include any new data
and therefore makes no statistical improvement to the
estimationprocedure. When windows are being applied however, the
overlap process can utilize data that wouldotherwise be
ignored.
The figure below shows data that is weighted with a Hanning
window. In this case the first and last20% of each sample period is
practically lost and contributes hardly anything towards the
averagingprocess.
Sampleddata
Processed datawith no overlap
Applying an overlap of at least 30% means that this data is once
again included - as shown below. Thisnot only speeds up the
acquisition (for the same number of averages) but also makes it
statistically
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more reliable since a much higher proportion of the acquired
data is being included in the averagingprocess.
Sampleddata
Processed data with 30% overlap
Aliasing
Sampling at too low a frequency can give rise to the problem of
aliasing which can lead to erroneousresults as illustrated
below.
This problem can be overcome by implementing what is known as
the Nyquist Criterion, whichstipulates that the sampling frequency
(fs) should be greater than twice the highest frequency of
theinterest (fm).
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The highest frequency that can be measured is fmax which is half
the sampling frequency (fs), and is alsoknown as the Nyquist
frequency (fn).
The problem of aliasing can also be illustrated in the frequency
domain.
inputfrequency
measuredfrequency
fn
fn 2 fn = fs
f1
f1 f2 f3 f43 fn 4 fn
All multiples of the Nyquist frequency (fn) act as folding
lines. So f4 is folded back on f3 around line 3 fn,f3 is folded
back on f2 around line 2 fn and f2 is folded back on f1 around line
fn. Therefore all signals atf2, f3, f4 are all seen as signals at
frequency f1.
The only sure way to avoid such problems is to apply an analog
or digital anti-aliasing filter to limit thehigh frequency content
of the signal. Filters are less than ideal however so the
positioning of the cut offfrequency of the filters must be made
with respect to fmax and the roll off characteristics of the
filter.
ideal filter
fmax fs
fmax fs
roll offcharacteristicsof a real filter
Leakage and windows
A further problem associated with the discrete time sampling of
the data is that of leakage. Acontinuous sine wave such as the one
shown below should result in the single spectral line.
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time
frequency
continuouswaveform
Because the signals are measured over a sample period T, the DFT
assumes that this is representativefor all time. When the sine wave
is not periodic in the sample time window, the result is a
consequentleakage of energy from the original line spectrum due to
the discontinuities at the edges.
The user should be aware that leakage is one of the most serious
problems associated with digitalsignal processing. Whilst aliasing
errors can be reduced by various techniques, leakage errors
cannever be eliminated. Leakage can be reduced by using different
excitation techniques and increasingthe frequency resolution, or
through the use of windows as described below.
Windows
The problem of discontinuities at the edge can be alleviated
either by ensuring that the signal and thesampling period are
synchronous or by ensuring that the function is zero at the start
and end of thesampling period. This latter situation can be
achieved by applying what is called a window functionwhich normally
takes the form of an amplitude modulated sine wave.
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Frequency spectrum ofa sine wave, periodic inthe sample period
T.
Frequency spectrum of asine wave, not periodicwith the sample
periodwithout a window.
Frequency spectrum ofa sine wave that is notperiodic with the
sampleperiod with a window.
sampleperiod T.
sampleperiod T.
X =
The use of windows gives rise to errors itself of which the user
should be aware and should be avoidedif possible. The various types
of windowing functions distribute the energy in different ways.
Thechoice of window depends on the input function and on your area
of interest.
Self windowing functionsSelf windowing functions are those that
are periodic in the sample period T or transient signals.Transient
signals are those where the function is naturally zero at the start
and end of the samplingperiod such as impulse and burst signals.
Self windowing functions should be adopted wheneverpossible since
the application of a window function presents problems of its own.
A rectangular oruniform window can then be used since it does not
affect the energy distribution.
Note! It should be noted that synchronizing the signal and the
sampling time, or using a selfwindowing function is preferable to
using a window.
Window characteristics
The time windows provided take a number of forms - many of which
are amplitude modulated sinewaves. There are all in effect filters
and the properties of the various windows can be compared
byexamining their filter characteristics in the frequency domain
where they can be characterized by thefactors shown below.
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The windows vary in the amount of energy squeezed in to the
central lobe as compared to that in theside lobes. The choice of
window depends on both the aim of the analysis and the type of
signal youare using. In general, the broader the noise Bandwidth,
the worse the frequency resolution, since itbecomes more difficult
to pick out adjacent frequencies with similar amplitudes. On the
other hand,selectivity (i.e. the ability to pick out a small
component next to a large on) is improved with side lobefalloff. It
is typical that a window that scores well on Bandwidth is weak on
side lobe fall off and thechoice is therefore a trade off between
the two. A summary of these characteristics of the windowsprovided
is given in Table 1.1.
Window type Highest sidelobe (dB)
Sidelobe falloff(dB/decade)
Noise Bandwidth(bins)
Max. Amperror (dB)
Uniform -13 -20 1.00 3.9
Hanning -32 -60 1.5 1.4
Hamming -43 -20 1.36 1.8
Kaiser-Bessel -69 -20 1.8 1.0
Blackman -92 -20 2.0 1.1
Flattop -93 0 3.43
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The following windows -Hanning, Hamming, Blackman, Kaiser-Bessel
andFlattop all take the form of an amplitude modulatedsine wave in
the time domain. For a comparison oftheir frequency domain filter
characteristics - seeTable 1.1.
HanningThis window is most commonly applied for general purpose
analysis of random signals with discretefrequency components. It
has the effect of applying a round topped filter. The ability to
distinguishbetween adjacent frequencies of similar amplitude is low
so it is not suitable for accuratemeasurements of small
signals.
HammingThis window has a higher side lobe than the Hanning but a
lower fall off rate and is best used when thedynamic range is about
50dB.
BlackmanThis window is useful for detecting a weak component in
the presence of a strong one.
Kaiser-BesselThe filter characteristics of this window provide
good selectivity, and thus make it suitable fordistinguishing
multiple tone signals with widely different levels. It can cause
more leakage than aHanning window when used with random
excitation.
FlattopThis windows name derives from its low ripple
characteristics in the filter pass band. This windowshould be used
for accurate amplitude measurements of single tone frequencies and
is best suited forcalibration purposes.
Force window
This type of window is used with a transient signal in the case
of impact testing.It is designed to eliminate stray noise inthe
excitation channel as illustrated here.It has a value of 1 during
the impact period and 0 otherwise.
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Exponential window
This window is also used with a transientsignal. It is designed
to ensure that the signal dies away sufficiently at the end of
thesampling period as shown below. Theform of the exponential
window is described by the formula e-bt . The `Exponential decay'
determines the % level at theend of the time window.
An exponential window is normally applied to the response
(output) channels during impact testing. It isalso the most
appropriate window to be used with a burst excitation signal in
which case it should beapplied to all channels i.e. force(s) and
response(s). It does however introduce artificial damping intothe
measurement data which should be carefully taken into account in
further processing in modalanalysis.
Choosing window functions
For the analysis of transient signals use :
Uniform for general purposes
Force for short impulses and transients to improve the signal to
noise ratio
Exponential for transients which are longer than the sample
period or which do notdecay sufficiently within this period.
For the analysis of continuous signals use :
Hanning for general purposes
Blackman orKaiser-Bessel
if selectivity is important and you need to distinguish between
harmonicsignals with very different levels
Flattop for calibration procedures and for those situations
where the correctamplitude measurements are important.
Uniform only when analyzing special sinusoids whose frequencies
coincide withcenter frequencies of the analysis.
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For system analysis i.e. measurement of FRFs use :
Window correction mode
Applying a window distorts the nature of the signal and
correction factors have to be applied tocompensate for this. This
correction can be applied in one of two ways.
Amplitude where the amplitude is corrected to the original
value.
Energy where the correction factor gives the correct signal
energy for a particular frequency band. Thisis the only method that
should be used for broad band analysis.
If a number of windows is applied to a function, the effect of
the window may be squared or cubed, andthis affects the correction
factor required.
Amplitude correctionConsider the example of a sine wave signal
and a Hanning window.
When the windowed signal (sine wave x Hanning window) is
transformed to the frequency domain, thenthe amplitude of the
resulting spectrum will be only half that of the equivalent
unwindowed signal. Thus
Force for the excitation (reference) signal when this is a
hammer
Exponential for the response signal of lightly damped systems
with hammer excitation
Hanning for reference and response channels when using random
excitationsignals
Uniform for reference and response channels when using pseudo
randomexcitation signals
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in order to correct for the effect of the Hanning window on the
amplitude of the frequency spectrum, theresulting spectrum has to
be multiplied by an amplitude correction factor of 2.
Amplitude correction must be used for amplitude measurements of
single tone frequencies if theanalysis is to yield correct
results.
Energy correctionWindowing also affects broadband signals.
In this case however it is the energy in the signal which it is
usually important to maintain, and anenergy correction factor will
be applied to restore the energy level of the windowed signal to
that of theoriginal signal.
In the case of a Hanning window, the energy in the windowed
signal is 61% of that the original signal.The windowed data needs
to be multiplied by 1.63 therefore to correct the energy level.
Window correction factors
The actual correction factor that is needed to compensate for
the application of the time windowdepends on the window correction
mode and the number of windows applied. Table 1.2 lists the
valuesused.
Window type Amplitude mode Energy mode
Uniform 1 1
Hanning x1 2 1.63
Hanning x2 2.67 1.91
Hanning x3 3.20 2.11
Blackman 2.80 1.97
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Hamming 1.85 1.59
Kaiser-Bessel 2.49 1.86
Flattop 4.18 2.26
Table 1.2 Window correction factors
Averaging
Signals in the real world are contaminated by noise -both random
and bias. This contamination can bereduced by averaging a number of
measurements in which the random noise signal will average tozero.
Bias errors however, such as nonlinearities, leakage and mass
loading are not reduced by theaveraging process. A number of
different techniques for averaging of measurements are
provided.
Linear
This produces a linearly weighted average in which all the
individual measurements have the sameinfluence on the final
averaged value. If the average value of M consecutive measurement
ensemblesis x then -
The intermediate average is xa?n=xan-1+xn. The final averaging
can be done at the end of theacquisition.
Stable
In the case of stable averaging again all the individual
measurements have the same influence on thefinal averaged value. In
this case though, the intermediate averaging result is based on
The advantage of stable averaging is that the intermediate
averaging results are always properlyscaled. This scaling however
makes the procedure slightly more time consuming.
Exponential
Exponential averaging on the other hand yields an averaging
result to which the newest measurementhas the largest influence
while the effect of the older ones is gradually diminished. In this
case -
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where t is a constant which acts as a weighting factor.
Peak level hold
In this case a comparison has to be made between individual
measurement ensembles. When theycontain complex data, comparison is
done based on the amplitude information. For peak level
holdaveraging, the last measurement ensemble consisting of k
individual samples, xn (k), (where k= 0...N-1and N is the
blocksize) is compared to the average of the n-1 previous steps,
xn-1(k).The new average xn(k), is then defined as
In this way, the averaging result contains, for a specific k,
the maximum value in an absolute sense ofall the ensembles,
considered during the averaging process.
Peak reference hold
In peak reference hold averaging, one channel determines the
averaging process. If xi is the ensemblefor channel i and xr
represents the reference channel, then the last measurement
ensemble xrn(k)(where k= 0...N-1) is compared to the average of the
n-1 previous steps, xrn-1(k).The new average xn(k), is then defined
as -
This way, the averaging result contains all values that coincide
with the maximum values for thereference channel.
Reading list
Signal and system theory
J. S. Bendat and A.G. Piersol.Random Data : Analysis and
Measurement ProceduresWiley - Interscience, 1971.
J. S. Bendat and A.G. Piersol.Engineering Applications of
Correlation and Spectral AnalysisWiley - Interscience, 1980.
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R.K. Otnes and L. Enochson.Applied Time Series AnalysisJohn
Wiley & Cons, 1978.
J. MaxMthodes et Techniques de Traitement du Signal (2
Tomes)Masson, 1972, 1986.
General literature in digital signal processing
A.V. Oppenheimer and R.W. SchaferDigital Signal
ProcessingPrentice Hall, Englewood Cliffs N.J., 1975.
L.R. Rabiner and B. GoldTheory and Application of Digital Signal
ProcessingPrentice Hall, Englewood Cliffs N.J., 1975.
K.G. Beauchamp and C.K. YueuDigital Methods for Signal
AnalysisGeorge Allen & Unwin, London 1979.
M. BellangerTraitement Numrique du SignalMasson, Paris 1981.
A. Peled and B. LiuDigital Signal ProcessingTheory, Design And
ImplementationJohn Wiley & Sons.
Discrete Fourier Transform
E.O. BrighamThe Fast Fourier TransformPrentice Hall, Englewood
Cliffs N.J., 1974.
R.W. RamirezThe FFT : Fundamentals and ConceptsPrentice Hall,
Englewood Cliffs N.J., 1985.
C.S. Burrus and T.W. ParksDFT/FFT and Convolution Algorithms :
Theory and ImplementationJohn Wiley & Sons, 1985.
H.J. NussbaumerFast Fourier Transform and Convolution
AlgorithmsSpringer Verlag, 1982.
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R.E. BlahutFast Algorithms for Digital Signal ProcessingAddison
Wesley, 1985.
IEEE-ASSP SocietyPrograms for Digital Signal ProcessingIEEE
Press, New York, 1979.
Digital signal processingDigital signal processingTime and
frequency domainsFourier transformThe Discrete Fourier Transform
(DFT)Hermitian symmetryThe Fast Fourier Transform
(FFT)BlocksizeFrequency rangeReal time BandwidthOverlap
AliasingLeakage and windowsWindowsWindowsSelf windowing
functions
Window characteristicsWindow typesUniform
windowHanningHammingBlackmanKaiser-BesselFlattopForce
windowExponential window
Choosing window functionsWindow correction modeAmplitude
correctionEnergy correction
Window correction factors
AveragingAveragingLinearStableExponentialPeak level holdPeak
reference hold
Reading listReading listSignal and system theoryGeneral
literature in digital signal processingDiscrete Fourier
Transform